Approximating Small Balanced Vertex Separators in Almost Linear Time

  • Sebastian BrandtEmail author
  • Roger Wattenhofer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


For a graph G with n vertices and m edges, we give a randomized Las Vegas algorithm that approximates a small balanced vertex separator of G in almost linear time. More precisely, we show the following, for any \(\frac{2}{3}\le \alpha <1\) and any \(0<\varepsilon <1-\alpha \): If G contains an \(\alpha \)-separator of size K, then our algorithm finds an \((\alpha +\varepsilon )\)-separator of size \(\mathcal O(\varepsilon ^{-1}K^2\log ^{1+o(1)} n)\) in time \(\mathcal O(\varepsilon ^{-1}K^3m\log ^{2+o(1)} n)\) w.h.p. In particular, if \(K\in \mathcal O({\text {polylog}}\, n)\), then we obtain an \((\alpha +\varepsilon )\)-separator of size \(\mathcal O(\varepsilon ^{-1}{\text {polylog}}\, n)\) in time \(\mathcal O(\varepsilon ^{-1}m\,{\text {polylog}}\, n)\) w.h.p. The presented algorithm does not require knowledge of K.

Due to space restrictions, no proofs are included in this version of the paper; the full version with a lot of additional material can be found at


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.ETH ZürichZürichSwitzerland

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